1. Chengfu Mu, Peking University
Evaluating the Phase Diagram at finite Isospin and Baryon Chemical Potentials in NJL model
Chengfu Mu, Peking University
Collaborated with Lianyi He , J.W.Goethe University
Prof. Yu-xin Liu, Peking University
Ref: Phys. Rev. D 82, (2010)

2. Outline Introduction: exotic pair states
Theoretical work: NJL Model at finite chemical potential
Phase diagram at finite chemical potential
Summary

3. Introduction
QCD at finite density possesses a rich phase structure:
1. Cold quark matter forms color superconductivity at high baryon density, for example, CFL, 2SC et al.
2. Pion superfluid occurs when the isospin chemical potential is larger than pion mass in vacuum. At low , the ground state is a superfluid pion condensate, but at high , it is a Fermi liquid with Cooper pairing. They are connected by BCS-BEC crossover.

4. BCS-BEC crossover crossover
Qijin Chen et. al.: Physics Reports 412, 1-88 (2005)
crossover
In the BCS and BEC regimes: only one excitation spectrum with a single component. For the crossover case the excitations have two distinct components.

5. In the standard BCS theory, the two fermionic species form a Cooper pair with same magnitude but opposite direction in momenta. But in real world, the two paired species have mismatched Fermi surface because of different chemical potential, unequal number densities or an external magnetic field et.al..
Clogston limit: upper limit of mismatch.
What is the ground state in the asymmetric matter?
This is a hot topic in both quark matter and condensed matter. Some candidates are proposed. We mainly discuss two of them: the Sarma phase and Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase.

6. The Breached Paired phase (also called Sarma phase)
Superfluid component is breached by normal one in the region where gapless excitations happen.
W.Yi and L.M. Duan: Phys. Rev. Lett. 97, (2006) SF BP
k1 k2

7. LOFF state is a spatially anisotropic ground state where the rotational symmetry and/or the translational symmetry are spontaneously broken. Each Cooper pair carries a total momentum
M.Alford et. al. PRD63, (2001)

8. Two Flavor Nambu–Jona-Lasinio Model
The lagrangian density :
Chiral condensate :
Pion condensates :
u and d quark potential :
Partition function :

9. The gap equations for m and :
The thermodynamic potential:
where
and
The gap equations for m and :

10. The explicit form of the gap equations:
BEC
BCS

11. Single-particle excitation gap:

12. Sarma phase at finite isospin and quark potential
From the conditions:
We obtain the gapless points:
where

13. Only in the case , there is the possibility to realize the Sarma phase
Only in the case , there is the possibility to realize the Sarma phase. There are 3 types:
Only the branch has two gapless nodes at
Only the branch has one gapless node at This is in our case.
The branch has one gapless node at , the branch has one gapless node at

14. The LOFF phase

15. where

16. The unphysical term from q does not affect the gap equations
Fukushima and Iida, PRD76, (2007);
J. O. Andersen and T. Brauner, PRD81, (2010).
The unphysical term from q does not affect the gap equations
for m and :
The optimal value of q obtained via minimizing

17. Phase Diagram:
Phase transition in the low isospin chemical potential

18. Chiral phase transition

19. Superuid-Normal Phase Transition and Tricritical Point

20. Gapless pion condensate and topological quantum phase transition

21. LOFF phase The phase diagram at large isospin chemical potential
1st
2nd
Normal SF
The LOFF window in the weak coupling case:
Our calculation indicates the LOFF window in the strong coupling region.

22. Quasiparticle dispersions in LOFF phase:
in the weak coupling case:

23. Quasiparticle dispersions in LOFF phase:
Three gapless nodes
Two blocking regions

24. Contour plot of the thermodynamic potential

26. 1st VI VI VI

27. Summary
In this work, we focus on the case with arbitrary isospin chemical potential and small baryon chemical potential
The phase diagram shows a rich phase structure since the system undergoes a crossover from a Bose-Einstein condensate of charged pions to a BCS superfluid with condensed quark–antiquark Cooper pairs when increases at , and a nonzero baryon chemical potential serves as a mismatch between the pairing species.
We observe a gapless pion condensation phase near the quadruple point The first order chiral phase transition becomes a smooth crossover when

28. At very large isospin chemical potential , an inhomogeneous LOFF superfluid phase appears in a window of , which should in principle exist for arbitrary large .
Between the gapless and the LOFF phases, the pion superfluid phase and the normal quark matter phase are connected by a first order phase transition. In the normal phase above the superfluid domain, we find that charged pions are still bound states even though becomes very large, which is quite different from that at finite temperature. Our phase diagram is in good agreement with that found in imbalanced cold atomic systems.
Thank you !